In chemistry, we have a set of symmetry operations (mirror planes, axis of rotation, screw planes, centers of inversion, etc.) that can be applied to molecules and/or crystal systems. Each of these symmetry operations has a corresponding linear algebra matrix that can be applied to the initial system matrix (ie. how the molecule is laid out in space). By applying the symmetry operation, it will often change the position of individual atoms within the molecule. If you work with these operations enough times, you can start to visualize both the physical and the mathematical relationships at the same time.

This concept is extremely important in single crystal x-ray crystallography. The diffraction pattern of the x-rays from the crystal is dependent upon how the crystal is packed. I have used this technique to help prove the existence of new molecules in certain synthesis reactions for my PhD.

Blimey , only the last sentence made sense to me. Did you get to name the molecules?

Yes, I did get to name them in a way. I used a shorthand form to nickname them rather than to follow the formal IUPAC naming scheme in my dissertation. In this case, the IUPAC name for each compound would be the length of a long sentence. My PhD committee members thanked me for using the short version in naming them.

In chemistry, we have a set of symmetry operations (mirror planes, axis of rotation, screw planes, centers of inversion, etc.) that can be applied to molecules and/or crystal systems. Each of these symmetry operations has a corresponding linear algebra matrix that can be applied to the initial system matrix (ie. how the molecule is laid out in space). By applying the symmetry operation, it will often change the position of individual atoms within the molecule. If you work with these operations enough times, you can start to visualize both the physical and the mathematical relationships at the same time.

This concept is extremely important in single crystal x-ray crystallography. The diffraction pattern of the x-rays from the crystal is dependent upon how the crystal is packed. I have used this technique to help prove the existence of new molecules in certain synthesis reactions for my PhD.

Blimey , only the last sentence made sense to me. Did you get to name the molecules?

Yes, I did get to name them in a way. I used a shorthand form to nickname them rather than to follow the formal IUPAC naming scheme in my dissertation. In this case, the IUPAC name for each compound would be the length of a long sentence. My PhD committee members thanked me for using the short version in naming them.

Awesome work dude , I just read a quick bit about the IUPAC name and didn't realise you couldn't just name your new molecule 'unobtainium' like you could with a new star - It makes logical sense actually

.... on a much more basic level, I work in a primary school where we use a learning aid called numicon. It has different shapes for each of the numbers, and you can visualise the shapes to add numbers together or subtract them. So if you put the 2 shape next to the 3 shape it makes the 5 shape. The most satisfying thing about it is the way that the different shapes fit together in the box so that they all make 10.

I wasted a lot of time in school wondering if squared numbers were more than silly abstractions until I realized that even painters and tile setters use them for estimates. Feynman insisted on maintaining some semblance of a mental model when formulating a problem and became famous for his symbolic diagrams.

Imagine wanting to make a truss out of non-uniform struts to curve and support various point loads. With a scale drawing of it, you can quickly determine the load on each strut by drawing a "funicular polygon" on top of it. The loads are drawn as vertical arrows from the load points, with a length proportional to the loads, and then lines parallel to the struts are drawn in a way that they can be measured to get the results.

Before there was Finite Element Analysis, I learned to get similar results just by imagining my artifact being made from half-cooked pasta. That makes it easy for me to imagine where it will bend and crack.

Far too few people realize that 10% error is usually close enough, and that all the information they choose to depend on the cloud for is absolutely useless to their imagination.